On multiple zeta values of extremal height
Masanobu Kaneko and Mika Sakata
Abstract
We give three identities involving multiple zeta values of height one and of maximal height; an explicit formula for the height-one multiple zeta values, a regularized sum formula, and a sum formula for the multiple zeta values of maximal height.
1 Main results
The multiple zeta value (MZV) is a real number given by the nested series ζ(k1, . . . , kr) = X
0<m1<···<mr
1 mk11· · ·mkrr
for each index set k = (k1, . . . , kr) of positive integers ki, with the last entry kr >1 for convergence. The quantities w(k) :=k1+· · ·+kr, d(k) :=r, and h(k) := #{i|ki >1,1≤ i≤r}are called respectively the weight, the depth, and the height of the index set k(or of the multiple zeta value ζ(k) =ζ(k1, . . . , kr)).
In this paper, we present the following three identities which involve multiple zeta values of extremal height, that is, the MZVs of height one or of maximal height (all components of the index are greater than one).
Theorem 1.1 (Explicit formula for the height-one MZV). For any integers r, k≥1, we have
ζ(1, . . . ,| {z }1
r−1
, k+ 1) =
min(r,k)X
j=1
(−1)j−1 X
w(a)=k,w(b)=r d(a)=d(b)=j
ζ(a+b), (1)
where, for two indices a = (a1, . . . , aj) and b = (b1, . . . , bj) of the same depth, ζ(a+b) denotes ζ(a1+b1, . . . , aj+bj).
Note that the right-hand side of this formula is symmetric in r and k, and thus the formula makes the dualityζ(1, . . . ,1
| {z }
r−1
, k+ 1) =ζ(1, . . . ,1
| {z }
k−1
, r+ 1) visible. (N.B. We use the duality in our proof, so that we are not giving an alternative proof of the duality.) To our knowledge, no such symmetric explicit formula for the height-one MZV has been known, except for the well-known symmetric generating function [1, 4]:
1− X
r,k≥1
ζ(1, . . . ,1
| {z }
r−1
, k+ 1)xryk = Γ(1−x)Γ(1−y)
Γ(1−x−y) = exp µX∞
n=2
ζ(n)xn+yn−(x+y)n n
¶ .
Also, we should remark that the right-hand side of the theorem is symmetric with respect to any permutations of the arguments, so that the theorem of Hoffman [5, Theorem 2.2]
ensures the right-hand side is a polynomial in the Riemann zeta valuesζ(n), the fact also can be seen from the generating function above. Moreover, we note that all the MZVs appearing on the right-hand side is of maximal height.
As a final remark, the case of r = 2 gives nothing but the “sum formula” for depth 2 (r = 1 gives the trivial identity ζ(k+ 2) = ζ(k+ 2)). It was H. Tsumura who first remarked that we could obtain the depth 2 sum formula if we looked at the behavior at s= 0 of the identity (3) in the next section for r= 2.
Recall the classical sum formula states that the sum of all MZVs of fixed weight and depth is equal to the Riemann zeta value of that weight. If we extend the sum to include non-convergent MZVs with the shuffle regularization, the result will be the height-one MZV (up to sign).
Theorem 1.2 (Shuffle-regularized sum formula). For any integers r, k≥1, we have X
w(k)=r+k d(k)=r
ζX(k) = (−1)r−1ζ(1, . . . ,| {z }1
r−1
, k+ 1),
where ζX(k) is the shuffle regularized value which will be recalled in §2.
We do not know if there exists any nice stuffle-regularized sum formula.
Finally, we give a kind of sum formula for the maximal-height MZVs in the form of generating function. This is essentially known, but may be new in this form of presentation.
Let T(k) be the sum of all multiple zeta values of weightkand of maximal height:
T(k) := X
k1+···+kr=k r≥1,∀ki≥2
ζ(k1, . . . , kr).
Recall the multiple zeta-star value ζ?(k1, . . . , kr) is given by the non-strict nested sum ζ?(k1, . . . , kr) = X
0<m1≤···≤mr
1 mk11· · ·mkrr
.
Theorem 1.3. We have the generating series identity 1 +
X∞ k=2
T(k)xk = µ
1 + X∞ n=1
ζ?(2, . . . ,2
| {z }
n
)x2n
¶µ 1 +
X∞ n=1
ζ(3, . . . ,3
| {z }
n
)x3n
¶ .
After some necessary preliminaries in the next section, we prove these results in§3.
2 Preliminaries
Recall the function introduced in [2], ξ(k1, . . . , kr;s) = 1
Γ(s) Z ∞
0
ts−1Lik1,...,kr(1−e−t)
et−1 dt, (2)
where Lik1,...,kr(z) is the multiple polylogarithm function defined by Lik1,...,kr(z) = X
0<m1<···<mr
zmr mk11· · ·mkrr
.
When kr > 1, the value at z = 1 of Lik1,...,kr(z) is nothing but the multiple zeta value ζ(k1, . . . , kr). The function ξ(k1, . . . , kr;s) is analytically continued to an entirefunction in s. In the special case where (k1, . . . , kr) = (1, . . . ,| {z }1
r−1
, k), Arakawa and the first-named author have established in [2, Theorem 8] the following identity (we interchange r and k and shift stos+ 1), which is crucial in our proofs of Theorems 1.1 and 1.2:
ξ(1, . . . ,| {z }1
k−1
, r;s+ 1) = (−1)r−1 X
a1+···+ar=k
∀ap≥0
µs+ar
ar
¶
ζ(a1+ 1, . . . , ar−1+ 1, ar+ 1 +s) (3)
+
r−2
X
i=0
(−1)iζ(1, . . . ,1
| {z }
k−1
, r−i)ζ(1, . . . ,1
| {z }
i
,1 +s),
for anyr, k≥1. Here, we have introduced a complex variablesin the outer-most exponent of the MZV;
ζ(k1, . . . , kr−1, kr+s) := X
0<m1<···<mr
1
mk11· · ·mkr−1r−1mkrr+s
.
As remarked in [7, Remark 3.7], equation (3) is equivalent to the connection formula of Euler’s type of the multi-polylogarithm Li1, . . . ,1
| {z }
k−1
,r(z). It is shown in [2] that the function ζ(k1, . . . , kr−1, kr+s) can be meromorphically continued to the wholes-plane, and has a pole at s= 0 if kr = 1. We need the description of the principal part ats= 0 in terms of regularized polynomials, which we now explain.
For an indexk= (k1, . . . , kr), we denote byZkX(T) and Zk∗(T) respectively the shuffle and the stuffle (harmonic) regularized polynomial associated to k. These are the polyno- mials in R[T] uniquely characterized by the asymptotics
Lik1,...,kr(z) = ZkX(−log(1−z)) +O((1−z)ε) asz→1 for some ε >0
and X
0<m1<···<mr<M
1 mk11· · ·mkrr
= Zk∗(logM+γ) +O(M−ε) asM → ∞ for someε >0, whereγ is Euler’s constant. We refer the reader to [6] for details about the regularizations.
We denote the constant termZkX(0) of the shuffle-regularized polynomialZkX(T) byζX(k) and call it the shuffle-regularized value of (possibly divergent) ζ(k). If k is of the form k= (k1, . . . , kn,1, . . . ,| {z }1
m
) withkn>1, m≥0, then bothZkX(T) andZk∗(T) are of degreem and each coefficient ofTi is a linear combination of multiple zeta values of weightm−i.
If m = 0 (and so n= r), then ZkX(T) = Zk∗(T) = ZkX(0) = Zk∗(0) = ζ(k1, . . . , kr). Now write
ZkX(T) = Xm i=0
ai(k)Ti
i! and Zk∗(T) = Xm i=0
bi(k)(T−γ)i
i! .
Then, as shown in [3], the principal parts at s= 0 of Γ(s+ 1)ζ(k1, . . . , kr−1, kr+s) and ζ(k1, . . . , kr−1, kr+s) are given respectively by
Γ(s+ 1)ζ(k1, . . . , kr−1, kr+s) = Xm
i=0
ai(k)
si +O(s) (s→0) (4)
and
ζ(k1, . . . , kr−1, kr+s) = Xm
i=0
bi(k)
si +O(s) (s→0). (5)
We take this opportunity to point out a flaw in the proof in [3]. The integral in the sum on the right of the equation below (32) may not converge. But the argument can easily be modified by splitting the integral R∞
0 on the left asR1
0 +R∞
1 and looking at the limits when s→0 separately.
3 Proofs
Proof of Theorem 1.1. Since we have the duality ζ(1, . . . ,| {z }1
r−1
, k+ 1) =ζ(1, . . . ,| {z }1
k−1
, r+ 1) and the right-hand side of (1) is symmetric inrandk, it is enough to prove the theorem under the assumption k ≥r. We proceed by induction on r. When r = 1, both sides become ζ(k+ 1) and the assertion is true for all k ≥ 1. Suppose r ≥2 and the theorem is true when the depth on the left is less than r (andk is greater than or equal to the depth).
We look at the values at s= 0 of both sides of (3). The value ξ(1, . . . ,| {z }1
k−1
, r; 1) on the left is evaluated in [2, Theorem 9] and is equal to ζ(1, . . . ,| {z }1
r−1
, k+ 1). Since the functions ζ(a1+ 1, . . . , ar−1+ 1, ar + 1 +s) with ar = 0 as well as ζ(1, . . . ,| {z }1
i
,1 +s) on the right have poles ats= 0, we need to look at the constant term of the Laurent expansion of the right-hand side. (Because ξ(1, . . . ,| {z }1
k−1
, r;s+ 1) is entire, all the poles on the right actually cancel out.) In what follows within the proof of Theorem 1.1, we simply write the constant term at s = 0 ofζ(k1, . . . , kr−1, kr+s) as ζ(k1, . . . , kr−1, kr) even when kr = 1, which is equal toZk∗
1,...,kr(γ) as recalled in the previous section. Note that these values satisfy the stuffle (harmonic) product rule. With this convention, we have
ζ(1, . . . ,| {z }1
r−1
, k+ 1) = (−1)r−1 X
a1+···+ar=k
∀ap≥0
ζ(a1+ 1, . . . , ar+ 1)
+ Xr−2
i=0
(−1)iζ(1, . . . ,| {z }1
k−1
, r−i)·ζ(1, . . . ,| {z }1
i+1
).
We apply the duality ζ(1, . . .1
| {z }
k−1
, r−i) =ζ(1, . . . ,1
| {z }
r−i−2
, k+ 1) in the second sum on the right
and use the induction hypothesis (sincer−i−1< r) to obtain ζ(1, . . . ,| {z }1
r−1
, k+ 1) = (−1)r−1 X
a1+···+ar=k
∀ap≥0
ζ(a1+ 1, . . . , ar+ 1)
+
r−2
X
i=0
(−1)i
r−Xi−1 j=1
(−1)j−1 X
w(a)=k,w(b)=r−i−1 d(a)=d(b)=j
ζ(a+b)·ζ(1, . . . ,1
| {z }
i+1
)
= (−1)r−1 X
a1+···+ar=k
∀ap≥0
ζ(a1+ 1, . . . , ar+ 1)
+
r−1
X
j=1
(−1)j−1 X
w(a)=k d(a)=j
r−Xj−1
i=0
(−1)i X
w(b)=r−i−1 d(b)=j
ζ(a+b)·ζ(1, . . . ,| {z }1
i+1
).
Now we expand the productζ(a+b)·ζ(1, . . . ,| {z }1
i+1
) by using the stuffle product and re-arrange the terms according to the number of 1’s to compute the inner sum
r−Xj−1
i=0
(−1)i X
w(b)=r−i−1 d(b)=j
ζ(a+b)·ζ(1, . . . ,1
| {z }
i+1
).
For that purpose, we introduce another notation. For a fixed index a = (a1, . . . , aj) of depth j and integersl, n≥0, we set
S(a, l, n) := X
w(b)=r−l d(b)=j,h(b)=n
ζ(a1+b1, . . . ,1, . . . , as+bs, . . . ,1, . . . , aj+bj),
where the sum runs over all b = (b1, . . . , bj) of weight r−l, depth j, and height n, and over all possible positions of exactly l 1’s in the arguments. Then, by the stuffle product rule, we have
X
w(b)=r−i−1 d(b)=j
ζ(a+b)·ζ(1, . . . ,| {z }1
i+1
) =
Xi+1
l=max(0,i+1−j)
Xj
n=i+1−l
µ n i+ 1−l
¶
S(a, l, n).
We note that, when we expand ζ(a+b)ζ(1, . . . ,| {z }1
i+1
) by the stuffle product, the number of 1’s in each term should at leasti+ 1−j whenj < i+ 1. And if the number of 1’s isl, then the heightnon the right varies fromi+ 1−ltoj. A particular term in the sum S(a, l, n) on the right comes in exactly ¡ n
i+1−l
¢ ways from the product ζ(a+b)ζ(1, . . . ,| {z }1
i+1
) on the left, because there are i+ 1−l out of n positions of the index a+b on the left which produces that particular term on the right by colliding i+ 1−l1’s at those positions.
When we sum this up alternatingly fori= 0, . . . , r−j−1 with signs, all coefficients of S(a, l, n) with n, l ≥1 vanish, because of the binomial identity Pn+l−1
i=l−1(−1)i¡ n
i+1−l
¢ = 0
if n, l≥1. Hence, also by the identity Pn−1
i=0(−1)i¡ n
i+1
¢= 1 if n≥1 (the case l = 0), we obtain
r−Xj−1
i=0
(−1)i X
w(b)=r−i−1 d(b)=j
ζ(a+b)·ζ(1, . . . ,| {z }1
i+1
) = Xj
n=1
S(a,0, n) + (−1)r−j−1S(a, r−j,0).
When j≤r−1, we havePj
n=1S(a,0, n) =P
w(b)=r,d(b)=jζ(a+b) and this gives
r−1X
j=1
(−1)j−1 X
w(a)=k,w(b)=r d(a)=d(b)=j
ζ(a+b). (6)
Finally, we have
r−1
X
j=1
(−1)j−1 X
w(a)=k d(a)=j
(−1)r−j−1S(a, r−j,0)
= (−1)r
r−1
X
j=1
X
w(a)=k d(a)=j
S(a, r−j,0)
= (−1)r X
a1+···+ar=k ap≥0,at least oneap=0
ζ(a1+ 1, . . . , ar+ 1).
Hence, this and the terms in
(−1)r−1 X
a1+···+ar=k
∀ap≥0
ζ(a1+ 1, . . . , ar+ 1)
with at least one ap = 0 cancel out, thereby remains the term (−1)r−1 X
w(a)=k,w(b)=r d(a)=d(b)=r
ζ(a+b). (7)
The sum of (6) and (7) gives the right-hand side of the theorem, and our proof is done.
Proof of Theorem 1.2. We multiply Γ(s+ 1) on both sides of the identity (3) and look at the constant terms of the Laurent expansions ats= 0. The left-hand side is holomorphic at s = 0 and gives the value ζ(1, . . . ,| {z }1
r−1
, k+ 1) as we already saw in the last subsection.
The function ¡s+ar
ar
¢Γ(s+ 1)ζ(a1+ 1, . . . , ar−1+ 1, ar+ 1 +s) on the right is holomorphic ats= 0 ifar>1, and in that case gives the valueζ(a1+ 1, . . . , ar−1+ 1, ar+ 1). Ifar= 0, then¡s+ar
ar
¢Γ(s+ 1)ζ(a1+ 1, . . . , ar−1+ 1, ar+ 1 +s) = Γ(s+ 1)ζ(a1+ 1, . . . , ar−1+ 1,1 +s) has a pole ats= 0 and its constant term of the Laurent expansion isζX(a1+ 1, . . . , ar+ 1) by (4). On the other hand, the function Γ(s+ 1)ζ(1, . . . ,| {z }1
i
,1 +s) has no constant term
at s = 0 because Z1, . . . ,X 1
| {z }
i+1
(T) = Ti+1/(i+ 1)!, and hence we conclude the proof of the theorem.
We remark that we can prove the theorem alternatively by computing directly the left-hand side using the regularization formula [6, (5.2)]. Also, by Theorem 1.2 and [6, Corollary 5], we easily obtain the following sum formula for the shuffle-regularized poly- nomials:
X
w(k)=r+k d(k)=r
ζX(k;T) =
r−1
X
i=0
(−1)r−1−iζ(1, . . . ,1
| {z }
r−1−i
, k+ 1)Ti i!
for any r, k ≥ 1, where ζX(k;T) = ZRX(w) in the notation of [6] with w being a word corresponding tok.
Proof of Theorem 1.3. This is almost obvious if we write ki (≥ 2) as ki = 2 +· · ·+ 2 (ki: even) or ki = 3 + 2 +· · ·+ 2 (ki: odd), and consider the stuffle product of ζ?(2, . . . ,2)ζ(3, . . . ,3) after writing ζ?(2, . . . ,2) as sums of ordinary multiple zeta values.
An alternative proof is given by using the main identity in [8]. As is already remarked there, if we specialize y= 0 and z=x2 in equation (3) in [8], we obtain
1 + X∞ k=2
T(k)xk= exp µX∞
n=1
ζ(2n) n x2n
¶
·exp µX∞
n=1
(−1)n−1ζ(3n) n x3n
¶ .
It is standard that exp
µX∞
n=1
ζ(2n) n x2n
¶
= Γ(1 +x)Γ(1−x) = Y∞ m=1
µ 1− x2
m2
¶−1
= 1 + X∞ n=1
ζ?(2, . . . ,2
| {z }
n
)x2n,
whereas the identity exp
µX∞
n=1
(−1)n−1ζ(3n) n x3n
¶
= 1 + X∞ n=1
ζ(3, . . . ,| {z }3
n
)x3n
is a special case of [6, Corollary 2 of Proposition 4].
Acknowledgements
The authors thank Hirofumi Tsumura for his suggestion to look at the identity (3) more closely, that lead us to Theorem 1.1. They also thank S. Yamamoto for his calling our attention to the reference [8] in relation to Theorem 1.3. This paper was written during the first author’s stay at the I.H.E.S., France. He thanks Francis Brown for the invitation and for all the hospitality and support given there. This work is supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B) 23340010 (M.K.) and Grant-in-Aid for JSPS Fellows 14J00005 (M.S.).
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